The LIL for canonical U-statistics of order 2

Abstract

Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, ${\mathrm{limsup}}_n(n\log \log n{)}^{-1}|{\sum }_{1<=i<j<=n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists $C<\infty$ such that, both, $E\min (h^2(X_1,X_2),u)<C\log \log u$ for all large u and $sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X){|}_2<1,\Vert g(X){\Vert }_2<1,\Vert f{\Vert }_{\infty }<\infty ,\Vert g{\Vert }_{\infty }<\infty \}<C$.

36 pages

References in corpus (2)

• Comparison of Sums of independent Identically Distributed Random Variables
• Necessary and Sufficient Conditions for the Strong Law of Large Numbers for U-statistics

Cited by in corpus (8)

• Exponential and moment inequalities for U-statistics
• Moment inequalities for U-statistics
• Tail behaviour of multiple random integrals and U-statistics
• The LIL for canonical $U$-statistics
• Quantile Estimation of A general Single-Index Model
• The LIL for $U$-statistics in Hilbert spaces
• Self-Normalized Moderate Deviations for Degenerate U-Statistics
• Weighted and multivariate Johnson--Schechtman inequalities with application to interpolation theory